I keep hearing about a scene that was in the UK airing, but not the US airing (and what about on the DVD version?)

Yes, but no, assuming we're talking about the 1996 TV movie (think so, but with the mixing of talk about that and the latest episode...).
There's at least four versions of that movie: the US tx version, the planned UK tx version, the re-edit of that which was done after the Dunblane massacre made the BBC iffy about shots of children (even teenagers) being machine-gunned, and if I recall right, slightly different re-edits on the VHS tape and the BBC1 tx version (because the BBC board and the BBFC asked for different cuts). Plus another version - which might be one of the above - was run in BBC2's Doctor Who night, and the various DVD releases have included subtly different versions including most of the above. There may be more due to early releases/screenings of cuts sent out before the above cuts had been made.
So there's too many variants for anyone to have ever had the patience to document (FX: starts counting, waiting for someone to post the linkto the site that does list them all!). But there's no major variations to them in terms of continuity stuff, unless you count the Master's dying words, or the mess-up over 12 lives/regenerations that was sorted out by a dedub.
Everything else was in the early alternative scripts. Which are enough to make a sane man hear an ever-lasting sound of drums. Tap-tap-tap-tap...

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